I was hatched in Lincoln within sight of Holy Memorial Stadium in the 9th year of The Bobfather, and lived there most of the 1970's. Still have a few relatives in the homeland. It was a much smaller and more sleepy town back then.
Pioneers Park is big and a person can spend a lot of time looking for birds in unproductive areas. Wilderness Park is a better bet. This long linear wooded park follows a creek and has good birding opportunities throughout. I prefer the southern end of the trail.
It's true that Pioneers Park is landscaped for human activities, and only has pockets of "nature" area. (And a nice view of downtown and the capitol, BTW) There's a small nature center with some fallow grassland, ponds and wooded areas around that. It was OK for woodpeckers, nuthatches, owls and hawks in the winter. Beale Slough was another place I watched the fairly common birds in the winter, since it's near my grandmother's place. I'll check into this other park you mention. Lincoln always had nice parks and neighborhoods, though not really much that could serve as wildlife habitat. I'd also like to check into the endangered Salt Creek Tiger Beetle while I'm there. Oh, and I did pick up a copy of Birding Nebraska in February, but personal recommendations specific to the season are usually more reliable than books.
Since you will have a day in late May in Lincoln, I would concentrate several sites. Wilderness Park at Pioneers Boulevard (south portion), WP at S. 14th parking lot, but on east side of the street into the park & across the old RR trail, or WP at the Saltillo Road parking lot. Other sites would be the NW end of Pawnee Lake at NW 126th and Superior; Jack Sinn WMA (roads poor if wet) along Ashland Road east of U.S. 77 south of Ceresco. This is on the Lancaster/Saunders County line. The north end of Wagontrain Lake SRA, near Hickman, can also be good.
I'll add those notes to my copy of Birding Nebraska. Thanks!
One difficult I'll face in Lincoln and especially in Minnesota is the large number of species. On Hawai'i there are very few species, many of which live in very specific locations/habitats, so it's usually pretty easy to ID the birds.